The control of stochastic systems is the focus of this research and it provides a study that has wide applications to many areas of science and engineering as well as applications within mathematics. Based on empirical data from a wide variety of physical phenomena, the family of stochastic processes called fractional Brownian motions is appropriate for stochastic models of physical phenomena. However most of the members of this family of fractional Brownian motions do not possess the Markovian property or the semimartingale property that are usually assumed for stochastic models. Since most controlled stochastic systems in continuous time have been described with a noise modeled by a Brownian motion there is demand to study control problems for stochastic systems with other fractional Brownian motions. These control problems require significantly different methods for analysis than the well developed methods for the control of systems with Brownian motions. These major differences have resulted in having few results available for the optimal control of linear systems driven by an arbitrary fractional Brownian motion. The investigators for this grant have initiated a major study of these control problems for linear systems. This study should not only provide results for an arbitrary fractional Brownian motion but also provide results for the control of linear systems driven by processes from a general family of square integrable continuous stochastic processes. This work will develop further the investigators? initial work on finite time horizon, quadratic cost control of linear stochastic systems. It is planned to expand this study to other cost functionals and to other types of systems, both finite and infinite dimensional. Specifically this work is planned to determine explicit expressions for the optimal control and the optimal cost for an infinite time horizon ergodic (or long run average) quadratic cost functional. Furthermore it is planned to study the control of linear stochastic partial differential equations for both finite and infinite time horizon control problems with a quadratic cost functional. The noise stochastic process and the control for these equations are allowed to be restricted to the boundary of the domain. Typically the controlled stochastic systems have unknown parameters so it is necessary to identify the parameters and control the system simultaneously. This class of problems is called adaptive control and the investigators plan to extend their initial work on adaptive control for a scalar linear system to multidimensional linear systems with an arbitrary fractional Brownian motion.
Many physical systems are controlled so the question of the determination of a best or optimal control arises. This area of research is called optimal control. Typically a mathematical model of a physical phenomena must account for perturbations of the system or unmodelled dynamics so a noise process is introduced in the model. These models are called stochastic systems. Given a controlled stochastic system and a cost for the use of control and the behavior of the system it is usually very difficult to obtain an explicit optimal control and the associated optimal cost. Furthermore the controlled stochastic systems in continuous time have been restricted to using one specific stochastic process, Brownian motion (white noise) as the noise. However empirical evidence from a wide variety of physical phenomena demonstrates a need for other noise processes, particularly for the family of fractional Brownian motions. The investigators for this grant have recently initiated a study of the control of linear systems with cost functionals that are quadratic in the system state and the control. They have obtained some results for explicit optimal controls for linear systems with an arbitrary fractional Brownian motion. These investigators plan to extend this work to more general linear systems and to infinite time horizon control problems with models that have an arbitrary fractional Brownian motion or a more general stochastic process. The optimal controls for these models can be used for control applications in many fields. The usefulness of fractional Brownian motions has been demonstrated in hydrology, telecommunications, turbulence, epilepsy and cognition as well as other areas. The optimal control results can be important in the use of effective controls for these physical phenomena. These results should have an important impact on many fields.